How to consider F to check if set V qualifies as a vector space or not if no information of field F is given? A set V qualifies to be a vector space if it satisfies properties under addition and scalar multiplication over a field F. Let's say if a set V is given but the field F is not specified and it is given to check if set V is a vector space or not. Shall I consider F as both R (set of real numbers) and C (set of complex numbers) to check if
V is a vector space or not OR anyone is sufficient??
 A: I'm going to interpret the question as being:

Let $V$ be an abelian group. When does there exist a field $F$ and an $F$-action on $V$ (i.e., scalar multiplication) which makes $V$ an $F$-vector space?

The first observation is that any field $F$ is an extension of $\mathbb Q$ or a finite field $\mathbb F_p$, depending on the characteristic. Thus, it suffices to consider whether $V$ is a $\mathbb Q$-vector space or a $\mathbb F_p$-vector space.
I claim $V$ can be made an $\mathbb F_p$-vector space iff $V$ is $p$-torsion. (i.e., for any $v\in V$, we have $pv=0$.)
$\Rightarrow$: $pv:=v+\dots+v\ (p\text{ times})=(1+\dots+1)v=0v=0$.
$\Leftarrow$: Any element $a\in\mathbb F_p\cong\mathbb Z/p$ lifts to a positive integer $\tilde a>0\in\mathbb Z$. Let $a\cdot v:=\tilde a\cdot v$. This is well-defined since $V$ is $p$-torsion. (Note that $\tilde a\cdot v$ simply means $v+\dots+v$, adding together $\tilde a$ times.) It is easily checked that this actually does make $V$ an $\mathbb F_p$-vector space.
Next, I claim $V$ can be made a $\mathbb Q$-vector space iff $V$ is uniquely divisible (i.e., for any $v\in V$ and nonzero integer $n$, there exists a unique $w\in V$ such that $v=nw$).
$\Rightarrow$: trivial.
$\Leftarrow$: For each $p/q\in\mathbb Q$ and $v\in V$, let $w\in V$ be such that $v=qw$. Now, we define $p/q\cdot v:=pw$. Again, it is easy to check that this action is well-defined, and that this makes $V$ a $\mathbb Q$-vector space.

To summarize, the answer is:

$V$ must either be uniquely divisible, or there must exist a prime number $p$ such that $V$ is $p$-torsion.

A: You can’t hope that considering $\mathbb R$ and $\mathbb C$ would be sufficient.
For example, consider the finite fields $\mathbb F_p$ or $\mathbb F_p \times \mathbb F_p$ where $p$ is a prime number.
On those « sets » $V$, you can define a scalar multiplication $\mathbb F_p \times V \to V$… just take the multiplication of the field!
But there is no obvious scalar multiplication $\mathbb R \times V \to V$. Hence considering $\mathbb R $ or $\mathbb C$ as « natural » fields for vector spaces can’t be done.
Note: if you’re doing physics, then vector spaces are in most of the cases used with $\mathbb R $ or $\mathbb C$ fields.
